Section 5: Square of Opposition , Barker, pp. 31-36. ***Read the section, then the following additional comments. Barker’s discussion pivots on the crucial distinction between the hypothetical and the existential viewpoints. Since the work of the nineteenth-century logician and mathematician George Boole, the hypothetical viewpoint has usually been adopted. Up until that time, in accord with what Aristotle taught, the existential viewpoint was assumed. The difference in a nutshell is this: Hypothetical Viewpoint: We cannot assume that everything referred to in our statements exists. Existential Viewpoint: We assume that everything referred to in our statements exists. Let’s say the baseball season has just started, and I say: All the players hitting fifty homeruns or more this year are pretty good sluggers. Now, undoubtedly, this is true as far as it goes, but there’s one problem: We don’t know if there are going to be any baseball players this year who are going to hit fifty or more homeruns. So, for right now, this is purely a hypothetical statement. Basically what I’m really saying is: If there should be any players hitting fifty or more homeruns , then they would be pretty good sluggers. Now let’s move ahead in time and assume that the baseball season is over, and there have been several players who hit fifty or more homeruns this season. Now if I make my statement, I’m referring to individuals who really exist, namely players who hit fifty or more home runs this year. Thus, in the terminology we are learning here, this is an existential statement; it is based on a category of things (baseball players) that exist . So, this is the difference between the hypothetical and the existential viewpoints. Under the hypothetical viewpoint we are saying something about a class of things which would be true if those things existed, but we don’t necessarily know that they do exist. Take the sentence: All hobbits live in middle earth. Is it true? Well, yes—that’s the nature of hobbits as J. R. Tolkien created them. But we don’t know that there are any hobbits. So, the statement can be made only from the hypothetical viewpoint, viz. 1 LESSON 3 Categorical Logic, Part 2

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If there were hobbits, then they would live in middle earth. Take another statement: No American who can run the 100 meter dash in under nine seconds has ever been on the Olympic team. Again the question is: Are there any Americans running that fast? (And, by the way, if you don’t know, that would be an incredible speed.) A lot depends on the answer. If there actually are no Americans who can run that fast, there wouldn’t be a whole lot of news in the statement. But if there are people who can bring off that feat, I, for one, would want to know: why aren’t those men or women getting on the team? Everything depends on whether we treat the statement

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